Stability Analyses of PP Tori

As has been summarized in an accompanying chapter — also see our related detailed notes — we have been trying to understand why unstable nonaxisymmetric eigenvectors have the shapes that they do in rotating toroidal configurations. For any azimuthal mode, <math>~m</math>, we are referring both to the radial dependence of the distortion amplitude, <math>~f_m(\varpi)</math>, and the radial dependence of the phase function, <math>~\phi_m(\varpi)</math> — the latter is what the Imamura and Hadley collaboration refer to as a "constant phase locus." Some old videos showing the development over time of various self-gravitating "constant phase loci" can be found here; these videos supplement the published work of Woodward, Tohline & Hachisu (1994).

Here, we focus specifically on instabilities that arise in so-called (non-self-gravitating) Papaloizou-Pringle tori and will draw heavily from two publications: (1) Papaloizou & Pringle (1987), MNRAS, 225, 267 — The dynamical stability of differentially rotating discs. III. — hereafter, PPIII — and (2) Blaes (1985), MNRAS, 216, 553 — Oscillations of slender tori.

PP III

Figure 2 extracted without modification from p. 274 of J. C. B. Papaloizou & J. E. Pringle (1987)

"The Dynamical Stability of Differentially Rotating Discs. III"

MNRAS, vol. 225, pp. 267-283 © The Royal Astronomical Society

Blaes85

Equilibrium Configuration

In our separate discussion of PP84, we showed that the equilibrium structure of a PP-torus is defined by the enthalpy distribution,

<math> H = \frac{GM_\mathrm{pt}}{\varpi_0} \biggl[ (\chi^2 + \zeta^2)^{-1/2} - \frac{1}{2}\chi^{-2} - C_B^' \biggr] . </math>

Normalizing this expression by the enthalpy at the "center" — i.e., at the pressure maximum — of the torus which, as we have already shown, is

<math> H_0 = \frac{GM_\mathrm{pt}}{2\varpi_0} [1-2C_B^' ] \, </math>

gives,

<math> [1-2C_B^' ]\biggl(\frac{H}{H_0}\biggr) = 2(\chi^2 + \zeta^2)^{-1/2} - \chi^{-2} -1 + [1 - 2C_B^' ]. </math>

Now, in our review of Kojima's (1986) work, we showed that the square of the Mach number at the "center" of the torus is,

<math>~\mathfrak{M}_0^2 \equiv \frac{(v_\varphi\|_0)^2}{(c_s\|_0)^2}</math>	<math>~=</math>	<math>~\frac{2(n+1)}{\gamma}\biggl[ \frac{1}{\chi_-} - 1 \biggr]^{-2}</math>
	<math>~=</math>	<math>~2n [ 1- 2C_B^' ]^{-1} </math>
<math>~\Rightarrow ~~~~ [1 - 2C_B^'] </math>	<math>~=</math>	<math>~\frac{2n}{\mathfrak{M}_0^2} \, , </math>

where, in obtaining this last expression we have related the adiabatic exponent to the polytropic index via the relation, <math>~\gamma = (n+1)/n</math>. Instead of specifying the system's Mach number, Blaes (1985) defines the dimensionless parameter,

<math>~\equiv</math>

<math>~\frac{2n}{\mathfrak{M}_0^2} \, .</math>

Implementing this parameter swap, the equilibrium expression becomes,

<math> \beta^2 \biggl(\frac{H}{H_0}\biggr) = 2(\chi^2 + \zeta^2)^{-1/2} - \chi^{-2} -1 + \beta^2 \, , </math>

or,

<math>~1 - \frac{1}{\beta^2}\biggl[\chi^{-2} - 2(\chi^2 + \zeta^2)^{-1/2} + 1 \biggr] \, .</math>

Looking at Figure 1 of Blaes85 — see also the coordinate definitions given in his equation (2.1) — I conclude that,

<math>~\chi = 1 - x\cos\theta</math> and <math>\zeta = x\sin\theta \, .</math>

Hence,

<math>~\frac{H}{H_0} </math>	<math>~=</math>	<math>~1 - \frac{1}{\beta^2}\biggl\{ [1 - x\cos\theta]^{-2} - 2[(1 - x\cos\theta)^2 + x^2\sin^2\theta]^{-1/2} + 1 \biggr\} </math>
	<math>~=</math>	<math>~1 - \frac{1}{\beta^2}\biggl\{ [1 - x\cos\theta]^{-2} - 2[(1 - 2x\cos\theta + x^2\cos^2\theta) + x^2(1-\cos^2\theta)]^{-1/2} + 1 \biggr\} </math>
	<math>~=</math>	<math>~1 - \frac{1}{\beta^2}\biggl[ (1 - x\cos\theta)^{-2} - 2(1 - 2x\cos\theta + x^2)^{-1/2} + 1 \biggr] \, .</math>

This matches equation (2.2) of Blaes85, if we acknowledge that Blaes uses <math>~f</math> — instead of the parameter notation, <math>~\Theta_H</math>, found in our discussion of equilibrium polytropic configurations — to denote the normalized enthalpy; that is,

<math>~f_\mathrm{Blaes85} = \Theta_H \equiv \frac{H}{H_0} \, .</math>

This expression for the enthalpy throughout a Papaloizou-Pringle torus is valid for tori of arbitrary thickness <math>~(0 < \beta < 1)</math>. When considering only slim tori, Blaes (1985) points out that this expression can be written in terms of the following power series in <math>~x</math> (see his equation 1.3):

<math>~\Theta_H</math>

<math>~1 - \frac{1}{\beta^2}\biggl[x^2 + x^3(3\cos\theta - \cos^3\theta) + \mathcal{O}(x^4) \biggr] \, .</math>

Blaes then adopts a related parameter that is constant on isobaric surfaces, namely,

<math>\eta^2 \equiv 1 - \Theta_H \, ,</math>

which is unity at the surface of the torus and which goes to zero at the cross-sectional center of the torus. Notice that <math>~\eta</math> tracks the "radial" coordinate that measures the distance from the center of the torus; in particular, keeping only the leading-order term in <math>~x</math>, there is a simple linear relationship between <math>~\eta</math> and <math>~x</math>, namely,

<math>~[1 - \Theta_H]^{1/2} \approx \frac{x}{\beta} \, .</math>

Cubic Equation Solution

For later use, let's invert the cubic relation to obtain a more broadly applicable <math>~x(\eta)</math> function. Because we are only interested in radial profiles in the equatorial plane — that is, only for the values of <math>~\theta = 0</math> or <math>~\theta=\pi</math> — the relation to be inverted is,

<math>~x^2 \pm 2 x^3</math>	<math>~=</math>	<math>~(\beta\eta)^2</math>
<math>~\Rightarrow ~~~~ x^3 \pm \tfrac{1}{2}x^2 \mp \tfrac{1}{2}(\beta\eta)^2</math>	<math>~=</math>	<math>~0 \, .</math>

Table 1: Example Parameter Values determined by iterative solution for <math>~\beta = \tfrac{1}{10}</math>
<math>~\eta</math>	<math>~\Gamma^2 = 54\beta^2\eta^2</math>	Inner solution <math>~(\theta = 0)</math> [Superior sign in cubic eq.]		Outer solution <math>~(\theta = \pi)</math> [Inferior sign in cubic eq.]
<math>~\eta</math>	<math>~\Gamma^2 = 54\beta^2\eta^2</math>	<math>^\dagger\biggl(\frac{x_\mathrm{root}}{\beta}\biggr)</math>	<math>~6(S + T)</math>	<math>^\dagger\biggl(\frac{x_\mathrm{root}}{\beta}\biggr)</math>	<math>~6(S + T)</math>
0.25	0.03375	0.244112	1.14647	0.256675	-0.84600
1.0	0.54	0.91909	1.55145	1.1378	-0.31732
^†Here, <math>~x_\mathrm{root}</math> has been determined via a brute-force, iterative technique.

Following Wolfram's discussion of the cubic formula, we should view this expression as a specific case of the general formula,

in which case, as is detailed in equations (54) - (56) of Wolfram's discussion of the cubic formula, the three roots of any cubic equation are:

<math>~x_1</math>	<math>~=</math>	<math>~ -\frac{1}{3}a_2 + (S + T) \, , </math>
<math>~x_2</math>	<math>~=</math>	<math>~ -\frac{1}{3}a_2 - \frac{1}{2} (S + T) + \frac{1}{2} \it{i} \sqrt{3} (S-T)\, , </math>
<math>~x_3</math>	<math>~=</math>	<math>~ -\frac{1}{3}a_2 - \frac{1}{2} (S + T) - \frac{1}{2} \it{i} \sqrt{3} (S-T)\, , </math>

where,

<math>~S</math>	<math>~\equiv</math>	<math>~[R + \sqrt{D}]^{1/3} \, ,</math>
<math>~T</math>	<math>~\equiv</math>	<math>~[R - \sqrt{D}]^{1/3} \, ,</math>
<math>~D</math>	<math>~\equiv</math>	<math>~Q^3 + R^2 \, ,</math>
<math>~Q</math>	<math>~\equiv</math>	<math>~\frac{3a_1 - a_2^2}{3^2} \, ,</math>
<math>~R</math>	<math>~\equiv</math>	<math>~\frac{3^2a_2 a_1 - 3^3a_0 - 2a_2^3}{2\cdot 3^3} \, . </math>

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Outer [inferior sign] Solution

Focusing, first, on the inferior sign convention, which corresponds to the "outer" solution <math>~(\theta = \pi)</math>, we see that the coefficients that lead to our specific cubic equation are:

<math>~a_2</math>	<math>~=</math>	<math>~- \tfrac{1}{2} \, ,</math>
<math>~a_1</math>	<math>~=</math>	<math>~0 \, ,</math>
<math>~a_0</math>	<math>~=</math>	<math>~\tfrac{1}{2}(\beta\eta)^2 \, .</math>

Applying Wolfram's definitions of the <math>~Q</math> and <math>~R</math> parameters to our particular problem gives,

<math>~Q</math>	<math>~\equiv</math>	<math>~\frac{3a_1 - a_2^2}{3^2} = -\biggl(\frac{a_2}{3}\biggr)^2 = - \frac{1}{2^2\cdot 3^2} \, ;</math>
<math>~R</math>	<math>~\equiv</math>	<math>~\frac{3^2a_2 a_1 - 3^3a_0 - 2a_2^3}{2\cdot 3^3} = \frac{1}{2\cdot 3^3} \biggl[ -\frac{ 3^3}{2}(\beta\eta)^2 + \frac{1}{2^2}\biggr] </math>
	<math>~\equiv</math>	<math>~\frac{1}{2^3\cdot 3^3} \biggl[ 1 - 2\cdot 3^3(\beta\eta)^2\biggr] \, . </math>

Defining the parameter,

<math>~\Gamma^2</math>

<math>~\equiv</math>

we therefore have,

<math>~(2\cdot 3)^6 D</math>	<math>~=</math>	<math>~( 1 - \Gamma^2 )^2-1 \, ,</math>
<math>~(2\cdot 3)^3S^3</math>	<math>~\equiv</math>	<math>~(2\cdot 3)^3 R + \sqrt{(2\cdot 3)^6D} </math>
	<math>~\equiv</math>	<math>~(1-\Gamma^2) + \sqrt{( 1 - \Gamma^2 )^2-1}</math>
	<math>~\equiv</math>	<math>~(1-\Gamma^2) + i\sqrt{1-( 1 - \Gamma^2 )^2} \, ;</math>
<math>~(2\cdot 3)^3T^3</math>	<math>~\equiv</math>	<math>~(1-\Gamma^2) - i\sqrt{1-( 1 - \Gamma^2 )^2} \, .</math>

ASIDE: The cube root of an imaginary number …

<math>~\ell^3</math>	<math>~=</math>	<math>~A \pm i \sqrt{1-A^2}</math>
	<math>~=</math>	<math>~r_\ell e^{i\theta_\ell} \, ,</math>

where,

and,

<math>~\theta_\ell</math>

<math>~\pm \tan^{-1}\biggl( \frac{\sqrt{1-A^2}}{A} \biggr) = \pm \cos^{-1}A \, .</math>

Now, according to this online resource, the three roots <math>~(j=0,1,2)</math> of <math>~\ell^3</math> are,

<math>~\ell_j = r_\ell^{1/3}e^{i(\theta_\ell + 2j\pi)/3)} \, ,</math>

which, for our specific problem gives,

<math>~e^{i\theta_\pm/3} \cdot e^{i(2j\pi/3)} \, ,</math>

where the subscript on <math>~\theta</math> refers to the <math>~\pm</math> in our original expression for <math>~\ell</math>.

In our particular case, after associating <math>~A \leftrightarrow (1-\Gamma^2)</math>, we can write,

<math>~ 2\cdot 3(S + T) </math>	<math>~=</math>	<math>~ \biggl[(1-\Gamma^2) + i\sqrt{1-( 1 - \Gamma^2 )^2} \biggr]^{1/3} + \biggl[(1-\Gamma^2) - i\sqrt{1-( 1 - \Gamma^2 )^2} \biggr]^{1/3} </math>
	<math>~=</math>	<math>~ e^{i\theta_+/3} \cdot e^{i(2j\pi/3)} + e^{i\theta_-/3} \cdot e^{i(2j\pi/3)} </math>
	<math>~=</math>	<math>~e^{i(2j\pi/3)} \biggl\{ e^{i[\cos^{-1}(1-\Gamma^2)]/3} + e^{-i[\cos^{-1}(1-\Gamma^2)]/3} \biggr\} </math>
	<math>~=</math>	<math>~e^{i(2j\pi/3)} \biggl\{\cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] + i\sin\biggl[ \tfrac{1}{3} \cos^{-1}(1-\Gamma^2)\biggr] </math>
		<math>~+ \cos\biggl[ \tfrac{1}{3} \cos^{-1}(1-\Gamma^2)\biggr] - i\sin\biggl[ \tfrac{1}{3} \cos^{-1}(1-\Gamma^2)\biggr] \biggr\} </math>
	<math>~=</math>	<math>~2 e^{i(2j\pi/3)} \cdot \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] \, .</math>

Similarly, we can write,

<math>~ 2\cdot 3(S - T) </math>	<math>~=</math>	<math>~ \biggl[(1-\Gamma^2) + i\sqrt{1-( 1 - \Gamma^2 )^2} \biggr]^{1/3} - \biggl[(1-\Gamma^2) - i\sqrt{1-( 1 - \Gamma^2 )^2} \biggr]^{1/3} </math>
	<math>~=</math>	<math>~ e^{i\theta_+/3} \cdot e^{i(2j\pi/3)} - e^{i\theta_-/3} \cdot e^{i(2j\pi/3)} </math>
	<math>~=</math>	<math>~e^{i(2j\pi/3)} \biggl\{ e^{i[\cos^{-1}(1-\Gamma^2)]/3} - e^{-i[\cos^{-1}(1-\Gamma^2)]/3} \biggr\} </math>
	<math>~=</math>	<math>~e^{i(2j\pi/3)} \biggl\{\cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] + i\sin\biggl[ \tfrac{1}{3} \cos^{-1}(1-\Gamma^2)\biggr] </math>
		<math>~- \cos\biggl[ \tfrac{1}{3} \cos^{-1}(1-\Gamma^2)\biggr] + i\sin\biggl[ \tfrac{1}{3} \cos^{-1}(1-\Gamma^2)\biggr] \biggr\} </math>
	<math>~=</math>	<math>~2i e^{i(2j\pi/3)} \cdot \sin\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] \, . </math>

Focusing specifically on the "j=0" root, and setting <math>~a_2 = -\tfrac{1}{2}</math>, we therefore have,

<math>~6x_1-1</math>	<math>~=</math>	<math>~ 6(S + T) </math>
	<math>~=</math>	<math>~2 \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] \, ;</math>
<math>~6x_2-1</math>	<math>~=</math>	<math>~ - \frac{1}{2} \biggl\{ 2 \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] - i\sqrt{3} \cdot 2i \sin\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] \biggr\} </math>
	<math>~=</math>	<math>~ -2\biggl\{ \frac{1}{2}\cdot \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] +\frac{\sqrt{3}}{2} \cdot \sin\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] \biggr\} </math>
	<math>~=</math>	<math>~ 2\biggl\{ \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2) + \frac{2\pi}{3}\biggr] \biggr\} \, ; </math>
<math>~6x_3-1</math>	<math>~=</math>	<math>~ - \frac{1}{2} \biggl\{ 2 \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] + i\sqrt{3} \cdot 2i \sin\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] \biggr\} </math>
	<math>~=</math>	<math>~ -2\biggl\{ \frac{1}{2}\cdot \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] -\frac{\sqrt{3}}{2} \cdot \sin\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] \biggr\} </math>
	<math>~=</math>	<math>~ 2\biggl\{ \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2) - \frac{2\pi}{3}\biggr] \biggr\} \, . </math>

Table 1: Analytically Evaluated Roots determined for <math>~\beta = \tfrac{1}{10}</math>
<math>~\eta</math>	<math>~\Gamma^2 = 54\beta^2\eta^2</math>	Inner solution <math>~(\theta = 0)</math> [Superior sign in cubic eq.]			Outer solution <math>~(\theta = \pi)</math> [Inferior sign in cubic eq.]
<math>~\eta</math>	<math>~\Gamma^2 = 54\beta^2\eta^2</math>	<math>~x_1/\beta</math>	<math>~x_2/\beta</math>	<math>~x_3/\beta</math>	<math>~x_1/\beta</math>	<math>~x_2/\beta</math>	<math>~x_3/\beta</math>
0.25	0.03375	-4.98744	0.24411	-0.25667	4.98744	-0.24411	0.25667
1.0	0.54	-4.78128	0.91909	-1.1378	4.78128	-0.91909	1.1378
		CONFIRMATION: In all cases, <math>~x^2 + 2x^3 = (\beta\eta)^2</math>			CONFIRMATION: In all cases, <math>~x^2 - 2x^3 = (\beta\eta)^2</math>

Inner [superior sign] Solution

Next, examing the superior sign convention, which corresponds to the "inner" solution <math>~(\theta = 0)</math>, we see that the coefficients that lead to our specific cubic equation are:

<math>~a_2</math>	<math>~=</math>	<math>~\tfrac{1}{2} \, ,</math>
<math>~a_1</math>	<math>~=</math>	<math>~0 \, ,</math>
<math>~a_0</math>	<math>~=</math>	<math>~- \tfrac{1}{2}(\beta\eta)^2 \, .</math>

Following the same set of steps that were followed in determining the "outer" solution, here we find: <math>~Q</math> remains the same; <math>~R</math> has the same magnitude, but changes sign; and, hence, <math>~D</math> remains the same. We therefore have,

<math>~(2\cdot 3)^3S^3</math>	<math>~=</math>	<math>~- (1-\Gamma^2) + i\sqrt{1-( 1 - \Gamma^2 )^2} \, ;</math>
<math>~(2\cdot 3)^3T^3</math>	<math>~=</math>	<math>~- (1-\Gamma^2) - i\sqrt{1-( 1 - \Gamma^2 )^2} \, ,</math>

which leads to the following expressions for the three "inner" roots:

<math>~6x_1+1</math>	<math>~=</math>	<math>~- 2 \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2)\biggr] \, ;</math>
<math>~6x_2+1</math>	<math>~=</math>	<math>~ - 2\biggl\{ \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2) + \frac{2\pi}{3}\biggr] \biggr\} \, ; </math>
<math>~6x_3+1</math>	<math>~=</math>	<math>~ - 2\biggl\{ \cos\biggl[ \tfrac{1}{3}\cos^{-1}(1-\Gamma^2) - \frac{2\pi}{3}\biggr] \biggr\} \, . </math>

Analytically Prescribed Eigenvector

Our Notation

As is explicitly defined in Figure 1 of our accompanying detailed notes, we have chosen to represent the spatial structure of an eigenfunction in the equatorial-plane of toroidal-like configurations via the expression,

<math>~\biggl[ \frac{\delta\rho}{\rho_0}\biggr]_\mathrm{spatial} </math>

<math>~\biggl\{ f_m(\varpi)e^{-im[\phi_m(\varpi)]} \biggr\} \, .</math>

Recognizing that the leading factor, <math>~f_m</math>, is, in general, composed of both a real part and an imaginary part, we can rewrite this expression as,

<math>~\biggl[ \frac{\delta\rho}{\rho_0}\biggr]_\mathrm{spatial} </math>	<math>~=</math>	<math>~\biggl[\mathrm{Re}(f_m) + i\mathrm{Im}(f_m)\biggr] \biggl[\cos(m\phi_m) - i\sin(m\phi_m) \biggr]</math>
	<math>~=</math>	<math>~ \biggl[\mathrm{Re}(f_m) \cos(m\phi_m) + \mathrm{Im}(f_m)\sin(m\phi_m)\biggr] + i \biggl[- \mathrm{Re}(f_m)\sin(m\phi_m) + \mathrm{Im}(f_m)\cos(m\phi_m)\biggr] </math>
	<math>~=</math>	<math>~\sqrt{[\mathrm{Re}(f_m)]^2 + [\mathrm{Im}(f_m)]^2} \biggl\{ \biggl[\frac{\mathrm{Re}(f_m) \cos(m\phi_m) + \mathrm{Im}(f_m)\sin(m\phi_m)}{\sqrt{[\mathrm{Re}(f_m)]^2 + [\mathrm{Im}(f_m)]^2} }\biggr] </math>
		<math>~ + i \biggl[\frac{- \mathrm{Re}(f_m)\sin(m\phi_m) + \mathrm{Im}(f_m)\cos(m\phi_m)}{\sqrt{[\mathrm{Re}(f_m)]^2 + [\mathrm{Im}(f_m)]^2} }\biggr] \biggr\} </math>
	<math>~=</math>	<math>~ \biggl\| \frac{\delta\rho}{\rho_0}\biggr\|_\mathrm{spatial} \biggl\{\sin(\beta_f + m\phi_m) + i \cos(\beta_f + m\phi_m)\biggr\} \, , </math>

where, the modulus of this function is,

<math>~\biggl\| \frac{\delta\rho}{\rho_0}\biggr\|_\mathrm{spatial} \equiv \sqrt{\biggl[ \frac{\delta\rho}{\rho_0}\biggr]\cdot\biggl[ \frac{\delta\rho}{\rho_0}\biggr]^*} </math>	<math>~=</math>	<math>~ \biggl\{ \biggl[\mathrm{Re}(f_m) \cos(m\phi_m) + \mathrm{Im}(f_m)\sin(m\phi_m)\biggr]^2 + \biggl[- \mathrm{Re}(f_m)\sin(m\phi_m) + \mathrm{Im}(f_m)\cos(m\phi_m)\biggr]^2 \biggr\}^{1/2} </math>
	<math>~=</math>	<math>~ \biggl\{ [\mathrm{Re}(f_m) \cos(m\phi_m)]^2 + 2\mathrm{Re}(f_m)\mathrm{Im}(f_m)\sin(m\phi_m)\cos(m\phi_m)+ [\mathrm{Im}(f_m)\sin(m\phi_m)]^2 </math>
		<math>~ + [\mathrm{Re}(f_m)\sin(m\phi_m)]^2 - 2\mathrm{Re}(f_m)\mathrm{Im}(f_m)\sin(m\phi_m)\cos(m\phi_m) + [\mathrm{Im}(f_m)\cos(m\phi_m)]^2 \biggr\}^{1/2} </math>
	<math>~=</math>	<math>~ \sqrt{ [\mathrm{Re}(f_m)]^2 + [\mathrm{Im}(f_m)]^2} \, , </math>

and where we have introduced a new angle, <math>~\beta_f</math>, such that,

<math>~\sin\beta_f = \frac{\mathrm{Re}(f_m)}{\sqrt{[\mathrm{Re}(f_m)]^2 + [\mathrm{Im}(f_m)]^2}}</math>	and	<math>~\cos\beta_f = \frac{\mathrm{Im}(f_m)}{\sqrt{[\mathrm{Re}(f_m)]^2 + [\mathrm{Im}(f_m)]^2}} </math>
<math>~\Rightarrow ~~~~ \beta_f </math>	<math>~\equiv</math>	<math>~\tan^{-1}\biggl[\frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr] \, .</math>

Alternatively, we could choose to write,

<math>~\biggl[ \frac{\delta\rho}{\rho_0}\biggr]_\mathrm{spatial} </math>

<math>~ \biggl| \frac{\delta\rho}{\rho_0}\biggr|_\mathrm{spatial} \biggl\{ \cos(\alpha_f - m\phi_m) + i \sin(\alpha_f - m\phi_m)\biggr\} = \biggl| \frac{\delta\rho}{\rho_0}\biggr|_\mathrm{spatial} e^{i(\alpha_f -m\phi_m)} \, , </math>

where we have introduced a new angle, <math>~\alpha_f</math>, such that,

<math>~\cos\alpha_f = \frac{\mathrm{Re}(f_m)}{\sqrt{[\mathrm{Re}(f_m)]^2 + [\mathrm{Im}(f_m)]^2}}</math>	and	<math>~\sin\alpha_f = \frac{\mathrm{Im}(f_m)}{\sqrt{[\mathrm{Re}(f_m)]^2 + [\mathrm{Im}(f_m)]^2}} </math>
<math>~\Rightarrow ~~~~ \alpha_f </math>	<math>~\equiv</math>	<math>~\tan^{-1}\biggl[\frac{\mathrm{Im}(f_m)}{\mathrm{Re}(f_m)} \biggr] \, .</math>

Notice that, when written in this form, it is clear from taking the complex conjugate of the function that,

<math>~\alpha_f - m\phi_m</math>	<math>~=</math>	<math>~0 </math>
<math>~\Rightarrow ~~~~ m\phi_m</math>	<math>~=</math>	<math>~\tan^{-1}\biggl[\frac{\mathrm{Im}(f_m)}{\mathrm{Re}(f_m)} \biggr] \, .</math>

General Formulation

From my initial focused reading of the analysis presented by Blaes (1985), I conclude that, in the infinitely slender torus case, unstable modes in PP tori exhibit eigenvectors of the form,

<math>~\frac{\delta W}{W_0} \equiv \biggl[ \frac{W(\eta,\theta)}{C} - 1 \biggr]e^{im\Omega_p t}e^{-y_2 (\Omega_0 t)} </math>

<math>~\biggl\{ f_m(\eta,\theta)e^{-i[m\phi_m(\varpi) - k\theta]} \biggr\} \, ,</math>

where we have written the perturbation amplitude in a manner that conforms with the notation that we have used in Figure 1 of a related, but more general discussion. [NOTE: Initially, I wrote "+ k" rather than "- k" in the exponent of the exponential term on the RHS of this expression; but experience shows that "- k" is required to achieve proper behavior of the "constant phase locus" plot, as displayed below.] As is summarized in §1.3 of Blaes (1985), for "thick" (but, actually, still quite thin) tori, "exactly one exponentially growing mode exists for each value of the azimuthal wavenumber <math>~m</math>," and its complex amplitude takes the following form (see his equation 1.10):

<math>~f_m(\eta,\theta)</math>

<math> ~\beta^2 m^2 \biggl[ 2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} \pm 4i\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta\cos\theta\biggr] + \mathcal{O}(\beta^3) \, . </math>

Aside from an arbitrary leading scale factor, we should therefore find that the amplitude (modulus) of the enthalpy perturbation is,

<math>~\biggl|\frac{\delta W}{W_0} \biggr|</math>

<math>~\sqrt{[\mathrm{Re}(f_m)]^2+ [\mathrm{Im}(f_m)]^2} \, ;</math>

and the associated phase function should be,

<math>~m\phi_m - k\theta</math>

<math>~\tan^{-1} \biggl\{ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr\} \, .</math>

[NOTE: Initially, I expected the argument inside the arctan function to be the ratio, <math>~\mathrm{Im}(f_m)/\mathrm{Re}(f_m)</math>; but experience shows that the reciprocal of this ratio is required to achieve proper behavior of the "constant phase locus" plot, as displayed below.]

Now, keeping in mind that, for the time being, we are only interested in examining the shape of the unstable eigenvector in the equatorial plane of the torus, we can set,

<math>~\cos\theta ~~ \rightarrow ~~ \pm 1 \, .</math>

Hence, we have,

<math>~\frac{1}{\beta^4 m^4}\biggl\|\frac{\delta W}{W_0} \biggr\|^2</math>	<math>~=</math>	<math>~\biggl[2\eta^2 - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2}\biggr]^2 + 16\biggl[\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta \biggr]^2 </math>
	<math>~=</math>	<math>~\frac{1}{[2^2(n+1)^2]^2}\biggl[2^3(n+1)^2\eta^2 - 3(n+1)\eta^2 - (4n+1) \biggr]^2 + \frac{2^3 \cdot 3\eta^2}{(n+1)} </math>
	<math>~=</math>	<math>~\frac{1}{[2^2(n+1)^2]^2}\biggl[(4n+1) - (n+1)[2^3(n+1)-3]\eta^2 \biggr]^2 + \frac{2^3 \cdot 3\eta^2}{(n+1)} </math>
<math>~\Rightarrow ~~~~ \biggl[\frac{2(n+1)}{\beta m} \biggr]^4 \biggl\|\frac{\delta W}{W_0} \biggr\|^2</math>	<math>~=</math>	<math>~\biggl[(4n+1) - (n+1)[2^3(n+1)-3]\eta^2 \biggr]^2 + 2^7 \cdot 3(n+1)^3\eta^2 \, . </math>

Also, keeping in mind that, because of the <math>~\cos\theta</math> factor, the sign on the imaginary term flips its sign when switching from the "inner" region to the "outer" region of the torus,

	<math>~m\phi_m</math>	<math>~=</math>	<math>~\tan^{-1}\biggl\{ \frac{(4n+1) - (n+1)[2^3(n+1)-3]\eta^2 }{2^{7/2}\cdot 3^{1/2}(n+1)^{3/2}\eta} \biggr\}</math>	over	inner <math>~(\theta=0)</math> region of the torus;
while
	<math>~m\phi_m</math>	<math>~=</math>	<math>~\tan^{-1}\biggl\{- \frac{(4n+1) - (n+1)[2^3(n+1)-3]\eta^2 }{2^{7/2}\cdot 3^{1/2}(n+1)^{3/2}\eta} \biggr\} + k\pi</math>	over	outer <math>~(\theta=\pi)</math> region of torus.

Incompressible Slim Tori

If we specifically consider geometrically slim, incompressible tori — that is, if we set the polytropic index, <math>~n=0</math> — to lowest order the eigenfunction derived by Blaes (1985) takes the form,

<math>~f_m(\eta,\theta)</math>

<math> ~\beta^2 m^2 \biggl[ 2\eta^2 \cos^2\theta - \frac{3\eta^2}{4} - \frac{1}{4} \pm 4i\biggl(\frac{3}{2}\biggr)^{1/2} \eta\cos\theta\biggr] + \cancelto{0}{\mathcal{O}(\beta^3)} \, . </math>

Goldreich, Goodman and Narayan (1986)

Unperturbed Slim Torus Structure

Goldreich, Goodman & Narayan (1986, MNRAS, 221, 339) — hereafter, GGN86 — also used analytic techniques to analyze the properties of unstable, nonaxisymmetric eigenmodes in Papaloizou-Pringle tori. They restricted their discussion to only the slimmest tori, so overlap between the GGN86 and Blaes85 work is easiest to recognize if we begin with the enthalpy distribution prescribed for a "slim torus" by Blaes (1985), as discussed above, namely,

<math>~H = H_0\Theta_H</math>

<math>~H_0 - \frac{H_0}{\beta^2}\biggl[r^2 + r^3(3\cos\theta - \cos^3\theta) + \mathcal{O}(r^4) \biggr] \, .</math>

[Note: Here we have replaced the variable name, <math>~x</math>, as used in Blaes85, with the variable name, <math>~r</math>, in order (1) to emphasize that the variable represents a dimensionless radial coordinate, and (2) to avoid conflict with the GGN86 variable, <math>~x</math>, which is a Cartesian coordinate with the standard dimension of length.]

Now, from our above discussion of equilibrium PP tori and recognizing that the Keplerian angular frequency at the location of the enthalpy maximum is,

<math>\Omega_K \equiv \frac{GM_\mathrm{pt}}{\varpi_0^3} \, ,</math>

we can set,

<math>~\frac{GM_\mathrm{pt}\beta^2}{2\varpi_0} = \tfrac{1}{2} \Omega_K^2 \varpi_0^2 \beta^2 \, .</math>

Hence, for the slimmest tori — that is, keeping only the lowest order term in <math>~r</math> — the enthalpy distribution becomes,

<math>~H </math>	<math>~=</math>	<math>~\tfrac{1}{2} \Omega_K^2 \varpi_0^2 \beta^2 - \tfrac{1}{2} \Omega_K^2 \varpi_0^2\biggl[r^2 + \cancelto{0}{r^3}(3\cos\theta - \cos^3\theta) + \cancelto{0}{\mathcal{O}(r^4)} ~~\biggr] </math>
	<math>~\approx</math>	<math>~\frac{\Omega_K^2}{2} [\varpi_0^2 \beta^2 - \varpi_0^2 r^2] \, .</math>

Following GGN86, the surface of the torus — where the enthalpy drops to zero — occurs at <math>~r = a/\varpi_0</math>. Hence, we recognize that,

<math>\beta = \frac{a}{\varpi_0} \, ,</math>

and we can rewrite the expression for the unperturbed enthalpy distribution as,

<math>~\frac{\Omega_K^2}{2} [a^2 - \varpi_0^2 r^2 ] \, .</math>

This expression exactly matches equation (2.13) of GGN86 — which, is,

<math>~\frac{\Omega^2}{2} \biggl[ (2q-3)(a^2 - x^2) - z^2 \biggr] \, ,</math>

once it is appreciated that, in moving from the Blaes85 discussion to the GGN86 discussion, <math>~\varpi_0^2 r^2 \rightarrow (x^2 + z^2)</math>, and it is recognized that Blaes85 restricted his investigation to tori that have uniform specific angular momentum <math>~(q = 2)</math>.

Additional Notation

<math>~(ky)_\mathrm{GGN} = \biggl( \frac{my}{\varpi_0} \biggr)_\mathrm{GGN} ~~\leftrightarrow ~~ (m\phi)_\mathrm{Blaes}</math>

<math>~\beta_\mathrm{GGN} \equiv \biggl( \frac{ma}{\varpi_0} \biggr)_\mathrm{GGN} ~~\leftrightarrow ~~ m\beta_\mathrm{Blaes}</math>

From equation (5.16) of GGN86 we obtain "the lowest order [complex] expression for the [perturbed] velocity potential," namely,

<math>~\tfrac{1}{4} k^2(5x^2 - 3z^2) \mp 4i\biggl(\frac{3}{2}\biggr)^{1/2} k x \beta_\mathrm{GGN} \, .</math>

Working on the imaginary part of this expression to put it in the terminology of Blaes85, we find,

<math>~\mathrm{Im}(\psi-1)</math>	<math>~=</math>	<math>~\mp 4\biggl(\frac{3}{2}\biggr)^{1/2} k x \beta_\mathrm{GGN} </math>
	<math>~=</math>	<math>~\mp 4\biggl(\frac{3}{2}\biggr)^{1/2} \biggl(\frac{m}{\varpi_0}\biggr) [\varpi_0 (\eta\beta_\mathrm{Blaes})\cos\theta ](m\beta_\mathrm{Blaes}) </math>
	<math>~=</math>	<math>~\mp 4\biggl(\frac{3}{2}\biggr)^{1/2} m^2\beta^2_\mathrm{Blaes} \eta\cos\theta \, ,</math>

which exactly matches <math>~\mathrm{Im}(f_m)</math> as derived by Blaes85 and summarized above. Similarly,

<math>~\mathrm{Re}(\psi-1)</math>	<math>~=</math>	<math>~\tfrac{1}{4} k^2(5x^2 - 3z^2) </math>
	<math>~=</math>	<math>~\frac{1}{4} \biggl(\frac{m}{\varpi_0}\biggr)^2[\varpi_0^2 r^2(5\cos^2\theta - 3\sin^2\theta)] </math>
	<math>~=</math>	<math>~\frac{1}{4} \eta^2 m^2 \beta^2_\mathrm{Blaes}[8\cos^2\theta - 3] </math>
	<math>~=</math>	<math>~m^2 \beta^2_\mathrm{Blaes}\biggl[2\eta^2\cos^2\theta - \frac{3\eta^2}{4}\biggr] \, .</math>

This exactly matches <math>~\mathrm{Re}(f_m)</math> as derived by Blaes85 and summarized above. This is in line with the following statement that appears in the acknowledgement section of GGN86: "We note that Omar Blaes … [has] independently derived many of the results reported in this paper."

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